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Simplifying (3x + 2)(4x + -5) + -1(2x + -1)(5x + 3) = 0 Reorder the terms: (2 + 3x)(4x + -5) + -1(2x + -1)(5x + 3) = 0 Reorder the terms: (2 + 3x)(-5 + 4x) + -1(2x + -1)(5x + 3) = 0 Multiply (2 + 3x) * (-5 + 4x) (2(-5 + 4x) + 3x * (-5 + 4x)) + -1(2x + -1)(5x + 3) = 0 ((-5 * 2 + 4x * 2) + 3x * (-5 + 4x)) + -1(2x + -1)(5x + 3) = 0 ((-10 + 8x) + 3x * (-5 + 4x)) + -1(2x + -1)(5x + 3) = 0 (-10 + 8x + (-5 * 3x + 4x * 3x)) + -1(2x + -1)(5x + 3) = 0 (-10 + 8x + (-15x + 12x2)) + -1(2x + -1)(5x + 3) = 0 Combine like terms: 8x + -15x = -7x (-10 + -7x + 12x2) + -1(2x + -1)(5x + 3) = 0 Reorder the terms: -10 + -7x + 12x2 + -1(-1 + 2x)(5x + 3) = 0 Reorder the terms: -10 + -7x + 12x2 + -1(-1 + 2x)(3 + 5x) = 0 Multiply (-1 + 2x) * (3 + 5x) -10 + -7x + 12x2 + -1(-1(3 + 5x) + 2x * (3 + 5x)) = 0 -10 + -7x + 12x2 + -1((3 * -1 + 5x * -1) + 2x * (3 + 5x)) = 0 -10 + -7x + 12x2 + -1((-3 + -5x) + 2x * (3 + 5x)) = 0 -10 + -7x + 12x2 + -1(-3 + -5x + (3 * 2x + 5x * 2x)) = 0 -10 + -7x + 12x2 + -1(-3 + -5x + (6x + 10x2)) = 0 Combine like terms: -5x + 6x = 1x -10 + -7x + 12x2 + -1(-3 + 1x + 10x2) = 0 -10 + -7x + 12x2 + (-3 * -1 + 1x * -1 + 10x2 * -1) = 0 -10 + -7x + 12x2 + (3 + -1x + -10x2) = 0 Reorder the terms: -10 + 3 + -7x + -1x + 12x2 + -10x2 = 0 Combine like terms: -10 + 3 = -7 -7 + -7x + -1x + 12x2 + -10x2 = 0 Combine like terms: -7x + -1x = -8x -7 + -8x + 12x2 + -10x2 = 0 Combine like terms: 12x2 + -10x2 = 2x2 -7 + -8x + 2x2 = 0 Solving -7 + -8x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + -4x + x2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -4x + 3.5 + x2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -4x + x2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -4x + x2 = 0 + 3.5 -4x + x2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -4x + x2 = 3.5 The x term is -4x. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4x + 4 + x2 = 3.5 + 4 Reorder the terms: 4 + -4x + x2 = 3.5 + 4 Combine like terms: 3.5 + 4 = 7.5 4 + -4x + x2 = 7.5 Factor a perfect square on the left side: (x + -2)(x + -2) = 7.5 Calculate the square root of the right side: 2.738612788 Break this problem into two subproblems by setting (x + -2) equal to 2.738612788 and -2.738612788.Subproblem 1
x + -2 = 2.738612788 Simplifying x + -2 = 2.738612788 Reorder the terms: -2 + x = 2.738612788 Solving -2 + x = 2.738612788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = 2.738612788 + 2 Combine like terms: -2 + 2 = 0 0 + x = 2.738612788 + 2 x = 2.738612788 + 2 Combine like terms: 2.738612788 + 2 = 4.738612788 x = 4.738612788 Simplifying x = 4.738612788Subproblem 2
x + -2 = -2.738612788 Simplifying x + -2 = -2.738612788 Reorder the terms: -2 + x = -2.738612788 Solving -2 + x = -2.738612788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = -2.738612788 + 2 Combine like terms: -2 + 2 = 0 0 + x = -2.738612788 + 2 x = -2.738612788 + 2 Combine like terms: -2.738612788 + 2 = -0.738612788 x = -0.738612788 Simplifying x = -0.738612788Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.738612788, -0.738612788}
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